Problem 29 - Brief Applied Calculus

Question

Evaluate each expression without using a calculator. $$ \left(\frac{27}{125}\right)^{2 / 3} $$

Step-by-Step Solution

Answer

The expression evaluates to \(\frac{9}{25}\).

Step 1: Understanding the Expression

We are given \[\left(\frac{27}{125}\right)^{2/3}\] This expression involves a fractional exponent, indicating both a power and a root. The base \(\frac{27}{125}\) is raised to the power of \(\frac{2}{3}\).

Step 2: Rewriting the Fraction and Exponent

The expression can be rewritten as:\[\left(\frac{27}{125}\right)^{2/3} = \left({27}^{1/3} \cdot {125}^{-1/3}\right)^2\]This structure shows that we will first take the cube roots of the numerator and denominator separately, then square the result.

Step 3: Taking the Cube Roots

Calculate the cube roots:* The cube root of 27 is 3 because 3 cubed is 27 (\(3^3 = 27\)).* The cube root of 125 is 5 because 5 cubed is 125 (\(5^3 = 125\)).So, we have:\[\left(\frac{27}{125}\right)^{1/3} = \frac{3}{5}\]

Step 4: Squaring the Result

Now, square the result of the cube roots:\[\left(\frac{3}{5}\right)^2 = \frac{3^2}{5^2} = \frac{9}{25}\]Thus, the squared result is \(\frac{9}{25}\).

Step 5: Final Answer

The value of the given expression is \(\frac{9}{25}\).

Key Concepts

Cube Roots

Cube roots are special because they tell us what number, when multiplied by itself three times, equals a given number. Unlike square roots, which are concerned with a pair of multipliers, cube roots deal with a trio. Understanding this concept is crucial when working with expressions like \( \left(\frac{27}{125}\right)^{2/3} \).

Let's break it down:

• For 27, the cube root is 3 because \( 3 \times 3 \times 3 = 27 \).
• For 125, the cube root is 5 since \( 5 \times 5 \times 5 = 125 \).

When you see a fractional exponent like \( \theta/3 \), the denominator tells you to take the cube root. This means each part of the fraction (numerator and denominator separately) needs to be analyzed in terms of cube roots before any other calculations.

Numerator and Denominator

The numerator and the denominator are the two key parts of a fraction. The numerator is the number on top, while the denominator is the number on the bottom. When given a fraction like \( \frac{27}{125} \), 27 is the numerator, and 125 is the denominator.

In expressions with fractional exponents, you work on the numerator and the denominator separately to simplify the operation. In our case:

• The numerator 27 is simplified by taking its cube root to get 3.
• The denominator 125 is simplified by taking its cube root to get 5.

This separation allows us to manage complex operations with ease, because once we have dealt with the roots, we can then move to more familiar calculations such as squaring.

Squaring Fractions

Squaring fractions can be simple if you remember that both the numerator and the denominator must be squared separately. When given a fractional outcome from cube roots, like the \( \frac{3}{5} \) from \( \left(\frac{27}{125}\right)^{1/3} \), the process of squaring follows these straightforward steps:

To square \( \frac{3}{5} \):

• Square the numerator: \( 3^2 = 9 \).
• Square the denominator: \( 5^2 = 25 \).

This gives you \( \frac{9}{25} \), the final answer of the original expression. Squaring a fraction separately ensures accuracy and keeps the calculations ordered. Always remember to apply the square to both parts of the fraction to avoid mistakes.